A227008 Numbers k such that Sum_{j=1..k} (sigma(j) + phi(j) + tau(j)) == 0 (mod k).

1, 399, 872, 1214, 2090, 5200, 5588, 21208, 29152, 638049, 1627676, 151732410, 274845607, 3224070252, 54892040166, 69020111756, 175288968221

Offset: 1

Author

Paolo P. Lava, Jun 27 2013

Keywords

• nonn
• more

Comments

a(17) > 10^11. - Donovan Johnson, Jul 07 2013

a(18) > 5*10^11. - Giovanni Resta, Jul 11 2013

Examples

			Sum_{j=1..399} sigma(j) = 130973;
Sum_{j=1..399} phi(j) = 48518;
Sum_{j=1..399} tau(j) = 2453;
(130973 + 48518 + 2453) / 399 = 456.
		

Crossrefs

Cf. A000005, A000010, A000203.

Cf. A006218, A002088, A024916.

Cf. A233541.

Programs

Maple

with(numtheory); ListA227008:=proc(q,h) local a, n;  a:=0;
for n from 1 to q do  a:=a+sigma(n)+phi(n)+tau(n); if (a mod n)=0 then print(n); fi; od; end: ListA227008(10^9);

PARI

s=0; for(n=1, 274845607, s=s+sigma(n)+eulerphi(n)+numdiv(n); if(s%n==0, print(n " " s))) /* Donovan Johnson, Jul 06 2013 */

Extensions

a(11)-a(14) from Donovan Johnson, Jul 06 2013

a(15)-a(16) from Donovan Johnson, Jul 07 2013

a(17) from Giovanni Resta, Jul 11 2013